Periodic at infinity functions of bounded variation

We consider functions of bounded variation defined on &$\mathbb {R}$ with their values in a complex Banachspace. We introduce the notions of slowly varying and periodic at infinity functions with bounded variation. The mainresults of the article are connected with harmonic analysis of periodic at infinity functions with bounded variation. For this class of functions we introduce the notion of a generalized Fourier series; the Fourier coefficients in this case may not be constants, they are functions that are slowly varying at infinity. We derive the analog of Wiener's theorem about absolutely convergent Fourier series for periodic at infinity functions with bounded variation. We also establish a criterion for representation of periodic at infinity functions as the sum of periodic functions and functions converging to zero. Basic results are derived with the use of isometric representations spectral theory.